Abstract
The paper is concerned with the problem of modelling the ensemble averaged product of the pressure and scalar gradient that appears in the equations for the scalar flux <u i θ>. The method (for nearly homogeneous unidirectional flow with weak gradients) is to derive nonlinear expressions for the fluctuating parts of the pressure and the scalar by formal solution (in wave vector space) of the Navier-Stokes and the scalar equations. Substitution in the Fourier transform of the pressure correlation shows that this quantity is the sum of four terms, one of which contains the mean scalar gradient. A fourth-order cumulant discard approximation allows this term to be expressed in terms of the single point double products and the turbulent energy and stress spectra. A numerical calculation is made when the spectra are represented by simple functions. Finally, a tentative model is proposed in which the remaining terms of the pressure correlation are evaluated by reference to experimental data.
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Abbreviations
- bij):
-
Anisotropy tensor 2<u i u j > /q 2 — 2δ ij /3
- Cθ 1 — Cθ 5):
-
Turbulence model constants
- I(r, t)):
-
Quantity defined by Eq. (7)
- G0(r,t;r1,t1)):
-
Green’s function, Eq. (10)
- k):
-
Wave vector
- kij):
-
j component of k i vector
- N(k,t)):
-
Fourier transform defined by Eq.(5)
- P):
-
Turbulent energy production rate
- p):
-
Fluctuating part of the pressure
- Pr t ):
-
Turbulent Prandtl number
- r):
-
Position vector
- Sij (k, t, t1)):
-
Two-time energy spectrum tensor
- Sij (k, t)):
-
One-time energy spectrum tensor
- T):
-
Mean value of scalar quantity
- t):
-
Time
- U):
-
Mean velocity
- u):
-
Velocity vector
- <uiuj>):
-
Reynolds stress
- <uiθ>):
-
Scalar flux
- V):
-
Volume
- v2 0):
-
\( \frac{1}{3}{q^{2}} \)
- q2 = <uiui>):
-
2 × turbulent kinetic energy
- xi (i = 1, 3)):
-
Cartesian coordinates
- α, β, γ):
-
Constants in spectrum models
- δ):
-
Dirac delta function
- δij):
-
Kronecker delta
- ε):
-
Turbulent energy dissipation rate
- θ):
-
Fluctuating part of scalar quantity
- λ):
-
Diffusivity
- ϱ):
-
Fluid density
- Фiθ):
-
Pressure-scalar-gradient correlation
- i, j, k):
-
Tensor indices
- *):
-
Complex conjugate
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© 1987 Springer-Verlag Berlin Heidelberg
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Dakos, T., Gibson, M.M. (1987). On Modelling the Pressure Terms of the Scalar Flux Equations. In: Durst, F., Launder, B.E., Lumley, J.L., Schmidt, F.W., Whitelaw, J.H. (eds) Turbulent Shear Flows 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-71435-1_2
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DOI: https://doi.org/10.1007/978-3-642-71435-1_2
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